Abstract
We refine the speed of convergence analysis for the quadratic augmented penalty algorithm. We improve the convergence order from 4/3 to 3/2 for the first order multiplier iteration. For the second order iteration, we generalize the analysis, and consider a primal–dual variant which asymptotically reduces to a Newton step for the optimality conditions.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Andreani, R., Birgin, E.G., Martínez, J.M., Schuverdt, M.L.: Augmented Lagrangian methods under the constant positive linear dependence constraint qualification. Math. Program. 111(1–2), 5–32 (2008)
Bertsekas, D.P.: Constrained Optimization and Lagrange Multiplier Methods. Academic Press, New York (1982)
Broyden, C.G., Attia, N.: A smooth sequential penalty function method for solving nonlinear programming problems. In: Thoft-Christenses, P. (ed.) System Modelling and Optimization, pp. 237–245. Springer, Berlin (1983)
Broyden, C.G., Attia, N.: Penalty functions, Newton’s method, and quadratic programming. J. Optim. Theory Appl. 58, 377–385 (1988)
Conn, A.R., Gould, N.I.M., Toint, P.L.: A globally convergent augmented Lagrangian algorithm for optimization with general constraints and simple bounds. SIAM J. Numer. Anal. 28(2), 545–572 (1991)
Dembo, R.S., Eisenstat, S.C., Steihaug, T.: Inexact Newton methods. SIAM J. Numer. Anal. 19(2), 400–408 (1982)
Dembo, R.S., Steihaug, T.: Truncated-Newton algorithms for large-scale unconstrained optimization. Math. Program. 26, 190–230 (1983)
Dolan, E.D., Moré, J.J.: Benchmarking optimization software with performance profiles. Math. Program. Ser. A 91, 201–213 (2002)
Dussault, J.-P.: Numerical stability and efficiency of penalty algorithms. SIAM J. Numer. Anal. 32(1), 296–317 (1995)
Dussault, J.-P.: Augmented penalty algorithms. IMA J. Numer. Anal. 18, 355–372 (1998)
Dussault, J.-P.: Augmented non quadratic penalty algorithms. Math. Program. 99(3), 467–488 (2004)
Fletcher, R.: Practical Methods of Optimization, 2nd edn. Wiley, New York (1987)
Gill, P.E., Murray, W., Wright, M.H.: Practical Optimization. Addison-Wesley, Reading (1978)
Gould, N.I.M.: On the accurate determination of search directions for simple differentiable penalty functions. IMA J. Numer. Anal. 6, 357–372 (1986)
Gould, N.I.M.: On the convergence of a sequential penalty function method for constrained minimization. SIAM J. Numer. Anal. 26, 107–108 (1989)
Gould, N.I.M., Orban, D., Toint, P.L.: CUTEr and SifDec: a constrained and unconstrained testing environment, revisited. ACM Trans. Math. Softw. 29(4), 373–394 (2003)
Hock, W., Schittkowski, K.: Test Examples for Nonlinear Programming Codes. Lecture Notes in Economics and Mathematical Systems, vol. 187. Springer, Berlin (1981)
Powell, M.J.D.: A method for nonlinear constraints in minimization problems. In: Fletcher, R. (ed.) Optimization. Academic Press, London (1969)
Scilab group: Ψlab 4.0. Institut National de Recherche en Informatique et Automatique, Domaine de Voluceau, Rocquencourt, France, e-mail: Scilab@inria.fr (2005)
Steer, S.: CUTEr. http://www.scilab.org/contrib/displayContribution.php?fileID=812 (2006)
Author information
Authors and Affiliations
Corresponding author
Additional information
This research was partially supported by NSERC grant OGP0005491.
Rights and permissions
About this article
Cite this article
Dussault, JP. Improved convergence order for augmented penalty algorithms. Comput Optim Appl 44, 373–383 (2009). https://doi.org/10.1007/s10589-007-9159-0
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10589-007-9159-0